As a related multi-party distributed multiplication device, there is a method which uses a system described in Non-Patent Document 1.
The method of Non-Patent Document 1 will be described hereinafter.
In Non-Patent Document 1, two arithmetic operation devices A, B are used, and values of a[1] and b[1] are inputted to one of the devices, A. Further, values of a[2] and b[2] are inputted to the other device B. The values a[1], b[1], a[2], and b[2] inputted to each of the devices A and B are set to be in relations of a[1]εZ/2Z, b[1]εZ/2Z, a[2]εZ/2Z, and b[2]εZ/2Z.
The two devices A and B of Non-Patent Document 1 execute arbitrary calculations in a distributed manner based on addition as well as multiplication of the inputted values (bits distributing to the two devices) by communicating with each other, and the device A outputs a value c[1] and the device B outputs a value c[2], respectively. The value c[1] outputted by the device A and the value c[2] outputted by the device B are in a relation satisfying c[1]+c[2]=(a[1]+a[2])(b[1]+b[2]). Moreover, the value c[1] and c[2] are set to be in relations of c[1]εZ/2Z and c[2]εZ/2Z. That is, the method of Non-Patent Document 1 is capable of distributing the product of a bit (a[1]+a[2]) and a bit (b[1]+b[2]) distributed to the respective devices A and B in a form of the sum again to the two devices A and B in a form of the sum as in c[1]+c[2].
In the meantime, in a case where the sum of the bit (a[1]+a[2]) and the bit (b[1]+b[2]) are distributed again to the respective devices A and B in a form of the sum as in c[1]+c[2], c[1] and c[2] turn out as c[1]=a[1]+b[1] and c[2]=a[2]+b[2], respectively, as a result of executing calculations in the two devices A and B in a variant manner. Thus, it is easy to distribute the sum of the bit (a[1]+a[2]) and the bit (b[1]+b[2]) again to the two devices A and B in a form of the sum as in c[1]+c[2].
As described above, according to Non-Patent Document 1, the use of two arithmetic operation devices make it possible to perform an arithmetic operation based on the bit values shared in the two devices. Thus, it is possible to employ distributed calculation processing according to Non-Patent Document 1 to arithmetic operation processing using a logic circuit. Further, since an arithmetic operation on a large ring can be written in a bit arithmetic operation, it is possible to employ the variant calculation processing of Non-Patent Document 1 to the arithmetic operation on the ring.    Non-Patent Document 1: Oded Goldreich: The Foundations of Cryptography—Volume 2. pp. 643-645 ISBN 0-521-83084-2 Published in US in May 2004, Publisher: Cambridge University Press